LPTMS PhD Proposal: Models and Time Series Analysis for Human Sports Performance

This project is directed to students with a strong background in quantitative methods from statistical physics, and ideally some knowledge of machine learning, computational physiology and statistical analysis of large data. Interest in sports performance would be useful. Expected are both analytical and computer programming
skills.

Models for human sports performances of various complexities and underlying principles have been proposed, often combining data from world record performances and bio-energetic facts of human physiology. For running, we were the first to derive an observed logarithmic scaling between world record running speeds and times from basic principles of metabolic power supply. We showed that various female and male record performances (world, national) and also personal best performances of individual runners for distances from 800m to the marathon are excellently described by our approach, with mean errors of (often much) less than 1%.

Main goal of this thesis project is the data-driven modeling of physiological and biomechanical processes in endurance sports, in particular running. The physiological and mechanical response of humans to exercise constitutes a complex system that involves many dynamical variables. Examples are the beat-to-beat intervals between heart beats, oxygen uptake, and stride frequency to name a few. These variables show inherent fluctuations that can be correlated.

Time series analysis can be used to detect these correlations which can show fractal scaling. This has been demonstrated for patients with cardiac diseases by Goldberger (see references below). Methods include detrended fluctuation analysis (DFA), multifractal DFA, EMD, multiscale entropy, and transfer entropy.

Models for complex physiological systems shall be constructed by learning from data. For example, running performance has been studied using recent advances in machine learning (see reference by Blythe and Kiraly). One aspect of this project is to apply machine learning to complex physiological data for endurance exercise and compare the so obtained results to findings from other methods.

LPTMS PhD Proposal: Amorphous materials under stress

Responsable:

Most of the materials employed in our everyday life have the amorphous structure of a collection of randomly packed particles. Their density and the strength of their interactions strongly depend on the material and are at the origin of a large spectrum of physical properties, much richer than the one of cristalline matter.
For example, under shear, gels and foams can flow as liquids, while glasses and ceramics break into solid pieces. Today we know that these behaviours are the macroscopic manifestations of genuine dynamical phase transitions anticipated by sudden reorganisations called avalanches. Avalanches are recorded at very different scales, from the micrometers of plastic readjustments to the kilometres of huge earthquakes. In analogy with what we know for equilibrium, these divergent scales indicate universality and strong fluctuations, but the dynamical transition of driven systems are much richer and mysterious due to effect of the preparation and to the temporal correlations.

In this thesis we will study the avalanche dynamics of amorphous matter using the tools developed in the context of disordered systems and the statistics of rare events. Connections with (many) body localisation physics will be explored.

Responsables:

During the last few years, it has been shown that non-interacting fermions in presence of a trap can be studied using the powerful techniques of random matrix theory, and more generally of determinantal point processes. This toolbox allowed to study analytically the interplay between quantum and thermal fluctuations in challenging physical situations where standard approaches of many-body physics, like the Local Density Approximation, fail.

Currently, there is an intense activity, both experimental and theoretical, concerning the relation between quantum entanglement and the thermodynamic entropy in many-body quantum systems. Many results remain currently conjectural. In this project, we would like to explore these questions, for instance the quantum entanglement at finite temperature, in a class of solvable models of trapped fermions, that are known to be related to RMT models at zero temperature.

LPTMS PhD Proposal: Extreme Value Statistics in Stochastic Processes

Responsables:

The extreme value statistics in stochastic processes is a subject of growing interest with applications from climate science to finance. Given a stochastic time-series over a given time interval [0,t], the typical questions are: what is the statistics of the maximum (or minimum) value of the process in this time window, at what time the maximum (or the minimum) is achieved, what is the time gap between the maximum and minimum etc.? Even for simple stochastic processes such as a one dimensional Brownian motion, these questions are often nontrivial. In the thesis, these questions would be addressed for a Brownian motion to start with, and progressively other types of stochastic processes would be studied.

LPTMS PhD Proposal: Mean field games

Mean field games present a new area of research at the boundary between applied mathematics,‭ ‬social sciences,‭ ‬engineering sciences and physics.‭ ‬It has been initiated a decade ago by Pierre-Louis Lions‭ (‬recipient of the‭ ‬94‭ ‬Fields medal‭) ‬and Jean-Michel Lasry as a new and promising tool to study many problem of social sciences,‭ ‬and with an explicit mention of the influence of concepts coming from physics‭ (‬the notion of‭ “‬mean field approximation‭”)‬.‭ ‬This field has since then grown significantly,‭ ‬and after a period where mainly stylized models where introduced,‭ ‬we witness now the appearance of‭ (‬necessarily more involved‭) ‬mean field game models closer to practical applications in finance,‭ ‬vaccination policies,‭ ‬or energy management through smart electronics.

Up to now,‭ ‬the development of Mean Field Games has mainly originated from the mathematics and economic communities.‭ ‬Mean Field Games theory is,‭ ‬however,‭ ‬by essence a multi-disciplinary field for which the input of physicists is much needed.‭ ‬Indeed,‭ ‬as important as they are,‭ ‬the studies of internal consistency and the numerical schemes developed by mathematicians cannot replace the deeper
understanding of the behavior of these models,‭ ‬obtained in particular through powerful approximation schemes,‭ ‬that physicists‭ (‬and essentially only them‭) ‬know how to provide.

For physicists a good‭ “‬entry point‭” ‬to the problematic of Mean Field Games is through the formal,‭ ‬but deep,‭ ‬connection between Mean Field Games and the nonlinear Schroedinger‭ (‬or Gross-Pitaevskii‭) ‬equation.‭ ‬This connection makes it possible to import to the field of Mean Field Games a variety of tools‭ (‬ranging from exact methods and approximation schemes to intuitive qualitative descriptions‭) ‬which have been developed along the year by physicists when studying interacting bosons or gravity waves in inviscid fluids.

The general subject of the proposed thesis is the study of Mean Field Games from a physicist point of view,‭ ‬that is with an objective to provide a true understanding‭ (‬through the identification of the relevant parameters and scale and the development of approximation schemes in the regimes of interest‭) ‬of the solutions of Mean Field Games equations.‭ ‬More specifically,‭ ‬two possible directions the proposed PhD ‬could take would be:

Self-organization is key to the function of living cells - but sometimes goes wrong! In Alzheimer's and many other diseases, normally soluble proteins thus clump up into pathological fiber-like aggregates. While biologists typically explain this on the grounds of detailed molecular interactions, we have started proving that such fibers are actually expected from very general physical principles. We thus show that geometrical frustration builds up when mismatched objects self-assemble, and leads to non-trivial aggregate morphologies, including fibers.

While our current numerical simulations let us monitor the aggregate formed by copies of a given irregular particle, we do not yet understand its dimensionality from first principles. Here we will tackle this problem using ideas inspired by the renormalization group. We will thus represent the shape of the particle as an orientation-dependent coupling matrix between two neighbors. By repeatedly applying a decimation procedure a la Kadanoff to the system, we will drastically reduce the enormous
space of all possible coupling matrices to a few fixed points corresponding to finite clusters, fibers, sheets and crystals. An internship will start with a transfer matrix calculation applying this procedure to a 1D system. A PhD will involve some non-trivial 3D discrete geometry and will likely require delving into
some linear algebra to understand the properties of our matrix universality classes.

Beyond protein aggregation, this project opens investigations into a new class of ''disordered'' systems where the disorder is carried by each identical particle instead of sprinkled throughout the system, and will help define the much-debated notion of frustration in dilute systems. This collaboration with Pierre Ronceray (Princeton U.) will ideally lead to collaborations with the experimental groups of Seth Fraden (Brandeis U.) and Friedrich Simmel (T.U. Munich).

Dissipation is ubiquitous in experiments on quantum matter and it typically reduces the timescales
over which pristine quantum phenomena can be investigated or lowers the quality of the
measurements. It’s an “enemy” that has to be fought harshly and roughly. In this internship we
will change the paradigm and consider dissipation as a resource. Dissipation can induce genuine
and interesting quantum effects (see for instance Ref. 1) and we are interesting in proposing
realistic experiments that can reveal them.

We will focus on the experiments on ultracold ytterbium gases that are currently realized in
several laboratories around the world, among which those at Collège de France in Paris (see
Ref. 2). The goal of this internship is to characterize theoretically the interplay between (i) the
dissipative mechanisms that distinguish these atoms and (ii) the unavoidable presence of atomatom
interactions (Ref. 3 presents some first data obtained in Hamburg, Germany). We will
inspect whether the dissipation-induced topological properties presented in the model of
reference 4, where interactions are neglected, can be observed in Ytterbium gases, where
interactions cannot be neglected. The main investigation tool will be advanced numerical
algorithms based on matrix-product states, that allow for the study of dissipative many-body
systems (see Ref. 5 for an article where such methods have been used to characterize dissipative
topological models).

Thesis proposal :
The quantum Hofstadter problem [1] (a charge particle hopping on a square lattice coupled to a perpendicular magnetic field) is fascinating for several reasons : its spectrum is a rare example of a fractal emerging from quantum mechanics, its transport properties can shed an interesting light on the quantum Hall effect,...
The Hofstadter model happens to be related to closed random walks on a square lattice,more precisely there is a mapping between the n-th moment of the Hofstadter Hamiltonian and the generating function for the enumeration of close lattice walks of length n enclosing a given algebraic area.
Recently [2] a formula for the algebraic area enumeration has been obtained starting from the so called Kreft coefficients [3] which encode the Schrodinger equation for the Hofstadter model.
Several key observations made in [2] and on which the enumeration relies happen to be incompletely understood and not yet seated on solid mathematical grounds.
Also the final enumeration formula has a complexity which increases exponentially with n making it difficult to be used for walks with a large number of steps.

The thesis will focus on a better understanding and improving of [2] namely :
i) derive some or all of the observations in [2] which lead to the enumeration formula
ii) interpret in terms of 1d random walks characteristics (if possible) the building blocks of the enumeration formula and use this interpretation to push further
its understanding
iii) also if possible simplify and reduce the formula to make it more tractable for the algebraic aera enumeration of walks with a large number of steps Last but not the least, coming back to the Hofstadter model via the mapping discussed above,can the enumeration formula gives new insights on the Hofstadter spectrum in the irrational limit where the flux per plaquette becomes an irrational number?

Last but not the least, coming back to the Hofstadter model via the mapping discussed above, can the enumeration formula gives new insights on the Hofstadter spectrum in the irrational limit where the flux per plaquette becomes an irrational number?

Correlated quantum systems in low dimensions show fascinating properties that distinguish them from their three dimensional counterparts as a consequence of the enhancement of quantum fluctuations. Interacting fermions and bosons in one-dimension (1D) can exhibit many exotic phases of matter. Although short-range interacting particles in 1D are rather well understood, much less is known for long-range interacting systems.Seminal efforts are underway in the control of artificial quantum systems to simulate arbitrary model Hamiltonians which are now barely accessible to classical computation methods. Ultra-cold dipolar or Rydberg atoms can realize Bose or Fermi gases with long-range interactions.